Rewriting terms as perfect derivative

Question

I am trying to find conservation laws of a system, and I'd like to rewrite terms of the form $$A\frac{dA^*}{dt}+A^*\frac{dA}{dt}$$

as $$\frac{d}{dt}|A|^2,$$ where $A$ is a complex valued function and $A^*$ is its conjugate.

Is it possible to take a large list of terms, and have mathematica rewrite all the terms it can as perfect derivatives?

EDIT: I will include a minimal example. I am trying to find a constant $\beta$ so that the terms provided below can be written in the form $\frac{\partial}{\partial X}\left(something\right)$:

(-A[X, T])*Ap[X, T]^2*Derivative[1, 0][A][X, T] - (5/2)*A[X, T]^2*Ap[X, T]^3*
Derivative[1, 0][A][X, T] - (1/2)*I*β*Ap[X, T]^2*Derivative[1, 0][A][X, 
T]^2 - 
A[X, T]^2*Ap[X, T]*Derivative[1, 0][Ap][X, T] - (5/2)*A[X, T]^3*Ap[X, T]^2*
Derivative[1, 0][Ap][X, T] + (5/2)*β*Ap[X, T]*Derivative[1, 0][A][X,
T]^2*
Derivative[1, 0][Ap][X, T] + (1/2)*I*β*A[X, T]^2*Derivative[1, 0][Ap][X, 
T]^2 + 
(5/2)*β*A[X, T]*Derivative[1, 0][A][X, T]*Derivative[1, 0][Ap][X, T]^2 - 
I*β*Ap[X, T]*Derivative[1, 0][A][X, T]*Derivative[1, 0][bϕ][X, T] + 
I*β*A[X, T]*Derivative[1, 0][Ap][X, T]*Derivative[1, 0][bϕ][X, T] - 
(1/4)*I*A[X, T]*Ap[X, T]^2*Derivative[2, 0][A][X, T] - 
(1/4)*β*Ap[X, T]^2*Derivative[1, 0][A][X, T]*Derivative[2, 0][A][X, T] + 
(1/2)*β*Derivative[1, 0][Ap][X, T]*Derivative[2, 0][A][X, T] + 
(3/2)*β*A[X, T]*Ap[X, T]*Derivative[1, 0][Ap][X, T]*
Derivative[2, 0][A][X, T] + 
(1/4)*I*A[X, T]^2*Ap[X, T]*Derivative[2, 0][Ap][X, T] + 
(1/2)*β*Derivative[1, 0][A][X, T]*Derivative[2, 0][Ap][X, T] + 
(3/2)*β*A[X, T]*Ap[X, T]*Derivative[1, 0][A][X, T]*
Derivative[2, 0][Ap][X, T] - 
(1/4)*β*A[X, T]^2*Derivative[1, 0][Ap][X, T]*Derivative[2, 0][Ap][X, T] 
+  
(1/8)*A[X, T]*Ap[X, T]^2*Derivative[3, 0][A][X, T] + 
(1/8)*I*β*Derivative[1, 0][Ap][X, T]*Derivative[3, 0][A][X, T] + 
(1/8)*A[X, T]^2*Ap[X, T]*Derivative[3, 0][Ap][X, T] - 
(1/8)*I*β*Derivative[1, 0][A][X, T]*Derivative[3, 0][Ap][X, T] - 
(1/16)*β*Derivative[1, 0][Ap][X, T]*Derivative[4, 0][A][X, T] - 
(1/16)*β*Derivative[1, 0][A][X, T]*Derivative[4, 0][Ap][X, T]

Answer 1

The approach below might give some ideas for how one can proceed.

expr = (-A[X, T])*Ap[X, T]^2*Derivative[1, 0][A][X, T] - (5/2)*
    A[X, T]^2*Ap[X, T]^3*Derivative[1, 0][A][X, T] - (1/2)*I*\[Beta]*
    Ap[X, T]^2*Derivative[1, 0][A][X, T]^2 - 
   A[X, T]^2*Ap[X, T]*Derivative[1, 0][Ap][X, T] - (5/2)*A[X, T]^3*
    Ap[X, T]^2*Derivative[1, 0][Ap][X, T] + (5/2)*\[Beta]*Ap[X, T]*
    Derivative[1, 0][A][X, T]^2*Derivative[1, 0][Ap][X, T] + (1/2)*
    I*\[Beta]*A[X, T]^2*Derivative[1, 0][Ap][X, T]^2 + (5/2)*\[Beta]*
    A[X, T]*Derivative[1, 0][A][X, T]*Derivative[1, 0][Ap][X, T]^2 - 
   I*\[Beta]*Ap[X, T]*Derivative[1, 0][A][X, T]*
    Derivative[1, 0][b\[Phi]][X, T] + 
   I*\[Beta]*A[X, T]*Derivative[1, 0][Ap][X, T]*
    Derivative[1, 0][b\[Phi]][X, T] - (1/4)*I*A[X, T]*Ap[X, T]^2*
    Derivative[2, 0][A][X, T] - (1/4)*\[Beta]*Ap[X, T]^2*
    Derivative[1, 0][A][X, T]*
    Derivative[2, 0][A][X, T] + (1/2)*\[Beta]*
    Derivative[1, 0][Ap][X, T]*
    Derivative[2, 0][A][X, T] + (3/2)*\[Beta]*A[X, T]*Ap[X, T]*
    Derivative[1, 0][Ap][X, T]*Derivative[2, 0][A][X, T] + (1/4)*I*
    A[X, T]^2*Ap[X, T]*Derivative[2, 0][Ap][X, T] + (1/2)*\[Beta]*
    Derivative[1, 0][A][X, T]*
    Derivative[2, 0][Ap][X, T] + (3/2)*\[Beta]*A[X, T]*Ap[X, T]*
    Derivative[1, 0][A][X, T]*
    Derivative[2, 0][Ap][X, T] - (1/4)*\[Beta]*A[X, T]^2*
    Derivative[1, 0][Ap][X, T]*Derivative[2, 0][Ap][X, T] +
   (1/8)*A[X, T]*Ap[X, T]^2*Derivative[3, 0][A][X, T] + (1/8)*
    I*\[Beta]*Derivative[1, 0][Ap][X, T]*
    Derivative[3, 0][A][X, T] + (1/8)*A[X, T]^2*Ap[X, T]*
    Derivative[3, 0][Ap][X, T] - (1/8)*I*\[Beta]*
    Derivative[1, 0][A][X, T]*
    Derivative[3, 0][Ap][X, T] - (1/16)*\[Beta]*
    Derivative[1, 0][Ap][X, T]*
    Derivative[4, 0][A][X, T] - (1/16)*\[Beta]*
    Derivative[1, 0][A][X, T]*Derivative[4, 0][Ap][X, T];

We set up the derivative relations in terms of a norm squared (I assume Ap is intended as the conjugate of A). I just use normA but it is the square really.

relation = A[X, T]*Ap[X, T] - normA[X, T];
replacements = NestList[D[#, X] &, relation, 4];

Find the variables we'll try to reduce.

mainvars = {A[X, T], Ap[X, T]};
allvars = Flatten[NestList[D[#, X] &, mainvars, 4]];

Compute a Groebner basis so we'll have a canonical reduction. Note that I use a term order that makes this fairly fast though it might not be best in terms of outcome.

gb = GroebnerBasis[replacements, allvars, 
   MonomialOrder -> DegreeReverseLexicographic, 
   CoefficientDomain -> RationalFunctions];

Now reduce modulo that Groebner basis.

PolynomialReduce[expr, gb, al(* {A[X, T]*Ap[X, T] - normA[X, T], 
Ap[X, T]*Derivative[1, 0][A][X, T] + 
 A[X, T]*Derivative[1, 0][Ap][X, T] - 
Derivative[1, 0][normA][X, T], 
2*Derivative[1, 0][A][X, T]*Derivative[1, 0][Ap][X, T] + 
 Ap[X, T]*Derivative[2, 0][A][X, T] + 
A[X, T]*Derivative[2, 0][Ap][X, T] - 
 Derivative[2, 0][normA][X, T], 
3*Derivative[1, 0][Ap][X, T]*Derivative[2, 0][A][X, T] + 
 3*Derivative[1, 0][A][X, T]*Derivative[2, 0][Ap][X, T] + 
 Ap[X, T]*Derivative[3, 0][A][X, T] + 
A[X, T]*Derivative[3, 0][Ap][X, T] - 
 Derivative[3, 0][normA][X, T], 
6*Derivative[2, 0][A][X, T]*Derivative[2, 0][Ap][X, T] + 
 4*Derivative[1, 0][Ap][X, T]*Derivative[3, 0][A][X, T] + 
 4*Derivative[1, 0][A][X, T]*Derivative[3, 0][Ap][X, T] + 
 Ap[X, T]*Derivative[4, 0][A][X, T] + 
A[X, T]*Derivative[4, 0][Ap][X, T] - 
 Derivative[4, 0][normA][X, T]} *)lvars, 
  MonomialOrder -> DegreeReverseLexicographic, 
  CoefficientDomain -> RationalFunctions][[2]]

(* {A[X, T]*Ap[X, T] - normA[X, T], 
 Ap[X, T]*Derivative[1, 0][A][X, T] + 
     A[X, T]*Derivative[1, 0][Ap][X, T] - 
  Derivative[1, 0][normA][X, T], 
   2*Derivative[1, 0][A][X, T]*Derivative[1, 0][Ap][X, T] + 
     Ap[X, T]*Derivative[2, 0][A][X, T] + 
  A[X, T]*Derivative[2, 0][Ap][X, T] - 
     Derivative[2, 0][normA][X, T], 
   3*Derivative[1, 0][Ap][X, T]*Derivative[2, 0][A][X, T] + 
     3*Derivative[1, 0][A][X, T]*Derivative[2, 0][Ap][X, T] + 
     Ap[X, T]*Derivative[3, 0][A][X, T] + 
  A[X, T]*Derivative[3, 0][Ap][X, T] - 
     Derivative[3, 0][normA][X, T], 
   6*Derivative[2, 0][A][X, T]*Derivative[2, 0][Ap][X, T] + 
     4*Derivative[1, 0][Ap][X, T]*Derivative[3, 0][A][X, T] + 
     4*Derivative[1, 0][A][X, T]*Derivative[3, 0][Ap][X, T] + 
     Ap[X, T]*Derivative[4, 0][A][X, T] + 
  A[X, T]*Derivative[4, 0][Ap][X, T] - 
     Derivative[4, 0][normA][X, T]} *)